Discounted robust control for Markov diffusion processes

In this paper we give conditions for the existence of discounted robust optimal policies under an infinite planning horizon for a general class of controlled diffusion processes. As for the attribute “robust” we mean the coexistence of unknown and non-observable parameters affecting the coefficients of the diffusion process. To obtain optimality, we rewrite the problem as a zero-sum game against nature, also known as worst case optimal control. Our analysis is based on the use of the dynamic programming technique by showing, among other facts, the existence of classical solutions (twice differentiable solutions) of the so-called Hamilton Jacobi Bellman equation. We provide an example on pollution accumulation control to illustrate our results.